# 6-4 Rhombuses, Rectangles and Squares Holt Geometry Warm Up Warm Up Lesson Presentation Lesson...

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6-4Rhombuses, Rectangles and SquaresHolt GeometryWarm UpLesson PresentationLesson Quiz

Warm UpSolve for x.

1. 16x 3 = 12x + 132. 2x 4 = 90

ABCD is a parallelogram. Find each measure.

3. CD4. mC447141046.4 Rhombuses, Rectangles and Squares

Prove and apply properties of rectangles, rhombuses, and squares.

Use properties of rectangles, rhombuses, and squares to solve problems.Objectives6.4 Rhombuses, Rectangles and Squares

rectanglerhombussquareVocabulary6.4 Rhombuses, Rectangles and Squares

A second type of special quadrilateral is a rectangle. A rectangle is a quadrilateral with four right angles.6.4 Rhombuses, Rectangles and Squares

Since a rectangle is a parallelogram by Theorem 6-4-1, a rectangle inherits all the properties of parallelograms that you learned in Lesson 6-2.6.4 Rhombuses, Rectangles and Squares

Example 1: Craft ApplicationA woodworker constructs a rectangular picture frame so that JK = 50 cm and JL = 86 cm. Find HM.Rect. diags. Def. of segs.Substitute and simplify. KM = JL = 866.4 Rhombuses, Rectangles and Squares

Check It Out! Example 1a Carpentry The rectangular gate has diagonal braces. Find HJ.Def. of segs.Rect. diags. HJ = GK = 486.4 Rhombuses, Rectangles and Squares

Check It Out! Example 1bCarpentry The rectangular gate has diagonal braces. Find HK.Def. of segs.Rect. diags. JL = LG JG = 2JL = 2(30.8) = 61.6 Substitute and simplify. Rect. diagonals bisect each other6.4 Rhombuses, Rectangles and Squares

A rhombus is another special quadrilateral. A rhombus is a quadrilateral with four congruent sides.6.4 Rhombuses, Rectangles and Squares

6.4 Rhombuses, Rectangles and Squares

Like a rectangle, a rhombus is a parallelogram. So you can apply the properties of parallelograms to rhombuses.6.4 Rhombuses, Rectangles and Squares

Example 2A: Using Properties of Rhombuses to Find MeasuresTVWX is a rhombus. Find TV. Def. of rhombusSubstitute given values.Subtract 3b from both sides and add 9 to both sides.Divide both sides by 10.WV = XT13b 9 = 3b + 410b = 13b = 1.36.4 Rhombuses, Rectangles and Squares

Example 2A ContinuedDef. of rhombusSubstitute 3b + 4 for XT.Substitute 1.3 for b and simplify.TV = XTTV = 3b + 4TV = 3(1.3) + 4 = 7.96.4 Rhombuses, Rectangles and Squares

Rhombus diag. Example 2B: Using Properties of Rhombuses to Find MeasuresTVWX is a rhombus. Find mVTZ. Substitute 14a + 20 for mVTZ.Subtract 20 from both sides and divide both sides by 14.mVZT = 9014a + 20 = 90a = 56.4 Rhombuses, Rectangles and Squares

Example 2B ContinuedRhombus each diag. bisects opp. s Substitute 5a 5 for mVTZ.Substitute 5 for a and simplify.mVTZ = mZTX mVTZ = (5a 5)mVTZ = [5(5) 5)] = 206.4 Rhombuses, Rectangles and Squares

Check It Out! Example 2a CDFG is a rhombus. Find CD.Def. of rhombusSubstituteSimplifySubstituteDef. of rhombusSubstituteCG = GF5a = 3a + 17a = 8.5GF = 3a + 17 = 42.5CD = GFCD = 42.56.4 Rhombuses, Rectangles and Squares

Check It Out! Example 2b CDFG is a rhombus. Find the measure.mGCH if mGCD = (b + 3)and mCDF = (6b 40)mGCD + mCDF = 180 b + 3 + 6b 40 = 180 7b = 217b = 31 Def. of rhombusSubstitute.Simplify.Divide both sides by 7.6.4 Rhombuses, Rectangles and Squares

Check It Out! Example 2b Continued mGCH + mHCD = mGCD2mGCH = mGCDRhombus each diag. bisects opp. s 2mGCH = (b + 3)2mGCH = (31 + 3)mGCH = 17Substitute.Substitute.Simplify and divide both sides by 2.6.4 Rhombuses, Rectangles and Squares

A square is a quadrilateral with four right angles and four congruent sides. In the exercises, you will show that a square is a parallelogram, a rectangle, and a rhombus. So a square has the properties of all three.6.4 Rhombuses, Rectangles and Squares

6.4 Rhombuses, Rectangles and Squares

Example 3: Verifying Properties of SquaresShow that the diagonals of square EFGH are congruent perpendicular bisectors of each other. 6.4 Rhombuses, Rectangles and Squares

Example 3 Continued6.4 Rhombuses, Rectangles and Squares

Example 3 Continued6.4 Rhombuses, Rectangles and Squares

The diagonals are congruent perpendicular bisectors of each other.Example 3 Continued6.4 Rhombuses, Rectangles and Squares

Check It Out! Example 3 The vertices of square STVW are S(5, 4), T(0, 2), V(6, 3) , and W(1, 9) . Show that the diagonals of square STVW are congruent perpendicular bisectors of each other.6.4 Rhombuses, Rectangles and Squares

Check It Out! Example 3 Continued 6.4 Rhombuses, Rectangles and Squares

Check It Out! Example 3 Continued 6.4 Rhombuses, Rectangles and Squares

The diagonals are congruent perpendicular bisectors of each other.Check It Out! Example 3 Continued 6.4 Rhombuses, Rectangles and Squares

Example 4: Using Properties of Special Parallelograms in ProofsProve: AEFD is a parallelogram.Given: ABCD is a rhombus. E is the midpoint of , and F is the midpoint of .6.4 Rhombuses, Rectangles and Squares

Example 4 Continued6.4 Rhombuses, Rectangles and Squares

Check It Out! Example 4 Given: PQTS is a rhombus with diagonalProve:6.4 Rhombuses, Rectangles and Squares

Check It Out! Example 4 Continued 1. PQTS is a rhombus.1. Given. 2. Rhombus eachdiag. bisects opp. s 3. QPR SPR 3. Def. of bisector.4. Def. of rhombus.5. Reflex. Prop. of 6. SAS7. CPCTC6.4 Rhombuses, Rectangles and Squares

StatementsReasons

Lesson Quiz: Part I A slab of concrete is poured with diagonal spacers. In rectangle CNRT, CN = 35 ft, and NT = 58 ft. Find each length.

1. TR2. CE

35 ft29 ft6.4 Rhombuses, Rectangles and Squares

Lesson Quiz: Part IIPQRS is a rhombus. Find each measure.

3. QP4. mQRP

42516.4 Rhombuses, Rectangles and Squares

Lesson Quiz: Part III5. The vertices of square ABCD are A(1, 3), B(3, 2), C(4, 4), and D(2, 5). Show that its diagonals are congruent perpendicular bisectors of each other.

6.4 Rhombuses, Rectangles and Squares

Lesson Quiz: Part IV6.4 Rhombuses, Rectangles and Squares

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